Shear-horizontal wave (SH wave)

A shear-horizontal wave propagating in the $x$ direction has displacement perpendicular to the sagittal $xy$ plane, for example $\mathbf{u}=w(x,y,t){\mathbf{e}}_z$. It is divergence-free and obeys the scalar shear-wave equation $w_{tt}=c_S^2(w_{xx}+w_{yy})$.

The SH input impedance is the negative ratio of vertical traction to particle velocity for a shear-horizontal wave field, with positive depth and tension-positive traction. A superposition of upward and downward waves generally has a depth-dependent complex ratio. A downward progressive wave in a uniform medium has $Z=\mu p_{\beta }$, while the upward wave has $Z=-\mu p_{\beta }$. The input ratio includes reflection from deeper structure and can have poles at velocity nodes.

With $e^{ikx-i\omega t}$ phasors, positive depth $z$, shear modulus $\mu =\rho {\beta }^2$, vertical traction $T$ and particle velocity $V$, the shear-wave equations give $V^{\prime }=-i\omega T/\mu$ and $T^{\prime }=-i\omega \mu p_{\beta }^{2}V$, where $p_{\beta }^2={\beta }^{-2}-k^2/{\omega }^2$. Define $Z=-T/V$ wherever $V\ne 0$. Differentiation gives the displayed Riccati equation and $V^{\prime }=i\omega ZV/\mu$. A pole of $Z$ can be an ordinary zero of $V$, not a singular physical wavefield.

In a uniform layer, put $q=\mu p_{\beta }$, $\kappa =\omega p_{\beta }$ and $d=z-z_r$. Initial impedance $Z_r$ gives the displayed fractional-linear solution and $V(z)=V_r[\cos \kappa d+i(Z_r/q)\sin \kappa d]$. Expanding into exponentials yields forward and backward shear-horizontal waves. Negative $d$ propagates a bottom load upward. If the denominator vanishes, the nonsingular pair $(V,T)$ must be propagated through the impedance pole.

Set the lower half-space load to $q_b={\mu }_b{p}_{\beta ,b}$ using the causal vertical wavenumber branch, which excludes a wave arriving from depth infinity. Apply the uniform-layer SH impedance update upwards through every layer, keeping particle velocity and traction continuous. For upper half-space impedance $q_a$, incident velocity $V_0$ and reflected velocity $R{V}_0$, the top values are $V=V_0(1+R)$ and $T=-q_a{V}_0(1-R)$. Their ratio gives the displayed reflection coefficient, incorporating every underlying interface and multiple reflection.

For positive real mass density and shear modulus, real $k$ and $\mathrm{Im}\omega >0$, the displayed derivative of $S_z=-\frac{1}{2}\mathrm{Re}(V^{\ast }T)$ follows from the two first-order shear equations. Every nontrivial solution has strictly decreasing $S_z$. Since $S_z=\frac{1}{2}\mathrm{Re}Z|V{|}^2$, positive-real impedance at a deeper point remains positive at all shallower points; negative-real impedance at a shallower point remains negative deeper down. A zero field is the necessary exception to strictness. At complex frequency this is a spectral flux diagnostic, reducing to a physical cycle average on the real axis.

For real horizontal wavenumber $k$, positive real wave speed $\beta$ and $\omega =u+iv$ with $v>0$, write $k_{\beta }=a+ib$ with $k_{\beta }^2={\omega }^2/{\beta }^2-k^2$. Its imaginary part gives $ab=uv/{\beta }^2$. The number $b$ cannot vanish, and $\mathrm{Re}(k_{\beta }/\omega )=v[b+u^2/({\beta }^{2}b)]/|\omega {|}^2$ has the sign of $b$. The branch $b>0$ makes $e^{i{k}_{\beta }z}$ decay towards increasing depth and analytically continues the outgoing propagating or evanescent branch to the upper frequency half-plane.

Between parallel boundaries, a guided shear-horizontal mode has a discrete transverse wavenumber $q_n$ and dispersion relation ${\omega }_n^2=c_S^2(k^2+q_n^2)$. Its cutoff frequency is $c_S{q}_n$.

A layer with shear modulus ${\mu }_1$ and wave speed $c_1$ lies between a rigid support and a half-space with shear modulus ${\mu }_2$ and wave speed $c_2>c_1$. A trapped antiplane shear normal mode has layer amplitude $A\sin (my)$ and exterior amplitude $A\sin (mh)e^{-\ell (y-h)}$, where $m=k\sqrt{c^2/c_1^2-1}$ and $\ell =k\sqrt{1-c^2/c_2^2}$. Continuity of displacement field and traction gives the displayed dispersion relation. It requires $c_1<c<c_2$ and differs from a Love wave with a free boundary.

For $A,ϵ>0$, consider $\tan \theta =-ϵ\theta /\sqrt{A^2-{\theta }^2}$ on $0<\theta <A$. On each negative branch of the tangent, the left side increases strictly, while the right side decreases strictly. A unique intersection exists precisely when the branch starts at $p_n=(n-1/2)\pi <A$, since the difference increases from negative infinity to a positive value or positive infinity. Consequently the number of trapped normal modes is $\lceil A/\pi -1/2\rceil$. The floor function expression $\lfloor A/\pi +1/2\rfloor$ agrees except at $A=p_n$, where it counts an inadmissible cutoff state with zero exterior exponential decay rate.